Editing Kronecker delta (section)

==Digital signal processing==
[[Image:unit impulse.gif|thumb|right|Unit sample function]]

In the study of [[digital signal processing]] (DSP), the Kronecker delta function sometimes means the unit sample function <math>\delta[n]</math> , which represents a special case of the 2-dimensional Kronecker delta function <math>\delta_{ij}</math> where the Kronecker indices include the number zero, and where one of the indices is zero:
<math display="block">\delta[n] \equiv \delta_{n0} \equiv \delta_{0n}~~~\text{where} -\infty<n<\infty</math>

Or more generally where:
<math display="block">\delta[n-k] \equiv \delta[k-n] \equiv \delta_{nk} \equiv \delta_{kn}\text{where} -\infty<n<\infty, -\infty<k<\infty</math>

For discrete-time signals, it is conventional to place a single integer index in square braces; in contrast the Kronecker delta, <math>\delta_{ij}</math>, can have any number of indexes. In [[LTI system]] theory, the discrete unit sample function is typically used as an input to a discrete-time system for determining the [[impulse response]] function of the system which characterizes the system for any general imput.  In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an [[Einstein summation convention]].

The discrete unit sample function is more simply defined as:
<math display="block">\delta[n] = \begin{cases} 1 & n = 0 \\ 0 & n \text{ is another integer}\end{cases}</math>

In comparison, in [[Discrete_time_and_continuous_time|continuous-time systems]] the [[Dirac delta function]] is often confused for both the Kronecker delta function and the unit sample function.  The Dirac delta is defined as:
<math display="block">\begin{cases} \int_{-\varepsilon}^{+\varepsilon}\delta(t)dt = 1 & \forall \varepsilon > 0 \\ \delta(t) = 0 & \forall t \neq 0\end{cases}</math>

Unlike the Kronecker delta function <math>\delta_{ij}</math> and the unit sample function <math>\delta[n]</math>, the Dirac delta function <math>\delta(t)</math> does not have an integer index, it has a single continuous non-integer value {{mvar|t}}.

In continuous-time systems, the term "[[unit impulse function]]" is used to refer to the [[Dirac delta function]] <math>\delta(t)</math> or, in discrete-time systems, the Kronecker delta function <math>\delta[n]</math>.